Complex hyperbolic (3, n, ∞) triangle groups

Let n >= 5 be a positive integer. A complex hyperbolic (3, n, infinity) triangle group is a representation from the hyperbolic (3, n, infinity) triangle group into the holomorphic isometry group of complex hyperbolic plane, which maps the generators to complex reflections fixing complex lines. In this paper, we show that a complex hyperbolic (3, n, infinity) triangle group < I-1, I-2, I-3 > is discrete and faithful if and only if I1I3I2I3 is not elliptic. Our result answers a conjecture of Schwartz for complex hyperbolic (3, n, infinity) triangle groups. (C) 2020 Elsevier Inc. All rights reserved.